When reading the book of Analysis of Financial Time Series, I have a question on the ARCH model, defined as follows
Regarding this model, the author also states that.
I do not understand why does the equation marked with yellow color is satisfied.
2026-03-26 06:16:36.1774505796
questions on a property of ARCH model
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$E(a_t^4|F_{t-1}) = E(\sigma_t^4\epsilon_t^4|F_{t-1}) = E(\epsilon_t^4|F_{t-1}) * E[(\sigma_t^2)^2|F_{t-1}] = 3(\alpha_0 + \alpha_1a_{t-1}^2)^2 $
I suggest reading up on the law of iterated expectations.
Ninja edit to your specific question is we know $E(\epsilon_t^4|F_{t-1})$ = 3